What changed: transition matrices, gains and losses

Two maps of the same landscape at two dates are the most common data of land change science, and the first question everyone asks of them is simple: what changed? In this tutorial you will compare the land use of Sibuyan Island, in the Philippines, in 1997 and in 2011. You will build the transition matrix that counts every cell's class at both dates, read from it each class's gain, loss, net change and swap, map where each transition happened, and split the total change into its quantity, exchange and shift components. At the end you will have a small model that does all of this for any pair of maps, the numbers a paper needs, and a sense of how much the answer depends on the size of the cells.

The idea#

A land use map is a raster: a grid of square cells, each holding a number that stands for a class (1 forest, 2 coconut plantation, 3 grassland...). When two rasters of the same place share the same grid (the same cells, the same size, the same position on the ground), every cell can be followed from the first date to the second. A cell that was forest in 1997 and grassland in 2011 has made the transition forest → grassland; a cell that was forest at both dates persisted.

Counting the cells of every pair of classes gives a square table, the cross-tabulation or transition matrix: one row per class at the first date, one column per class at the second. The diagonal holds the persistence; everything off the diagonal is change. Here is a toy landscape of 25 cells at two times, with three classes: forest (F), farming (A) and grassland (G).

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Its transition matrix, as change.crosstab gives it (rows: time 1; columns: time 2):

Everything a change analysis reports can be read from such a table. Robert Gilmore Pontius Jr. and colleagues made this explicit in a paper that changed how land change is reported (Pontius et al., 2004). Most land in most landscapes does not change in a decade, so the persistence on the diagonal dominates the matrix and hides the transitions that matter. They proposed to describe each class by four numbers taken from its row and column:

In the toy, forest gained nothing and lost 4 cells: a net change of 4 and no swap. Farming gained 3 and lost 1: a net change of 2 and a swap of 2. Grassland did the same. Over the whole landscape, the net change is the sum of the classes' net changes divided by two (every cell lost by one class is gained by another), and the same holds for the swap: 4 cells of net change and 2 of swap, 6 cells of change in all.

Pontius and Santacruz (2014) later split the swap further, in the R package diffeR. Part of a swap is an exchange: a pair of classes trading cells, one cell of farming becoming grassland while another of grassland becomes farming. The rest is shift: a class that moves by gaining from one class and losing to another (in the Sibuyan maps below, grassland gains from forest and loses to coconut). The net change is called the quantity component.

The toy's 6 cells of change are 4 of quantity, 2 of exchange and none of shift. Terraflux's change.components computes these numbers exactly as diffeR does.

When to use it#

Cross-tabulation is the first step of nearly every study of land change, and often the only one needed:

• Describing change. How much forest was lost between two censuses, to what, and where? Which classes grew? A transition matrix with the gains and losses answers it and is the table reviewers expect.
• Preparing a simulation. Land change models (Dinamica EGO, CLUE-S, the Markov chain) are calibrated on the transitions observed between two dates. change.suggest ranks them and suggests those large enough to model.
• Comparing two maps of the same date. The same table compares a map with a reference (a simulated map with the observed one, a classification with field data). Then the rows are the map being judged and the columns the reference, and gain and loss become omission and commission (see part 3).

What it is not for: a transition matrix counts cells, it says nothing about how fast the change was against the time it took or whether a transition targeted a class more than its size would explain. That is the intensity analysis of part 2. It also treats every cell alike, so a forest that lost its core and a forest whose edge moved by one cell look the same; GuidosToolbox's morphological change detection, in part 4, tells them apart. And it does not tell real change from map error: if the maps are 90% accurate, most of a 10% change could be error (Olofsson et al., 2014).

What you need#

Two maps of classes on the same grid, made with the same legend and, ideally, the same method. The file used here, sibuyan_landuse.tif, is part of the example data of Terraflux's library: Sibuyan Island's land use in 1997 and 2011, from the demonstration data of the CLUE-S model (Verburg et al., 2002), modified for teaching: use it to learn, not to draw conclusions about Sibuyan.

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Drag the handle to see what changed:

1997

2011

A coordinate reference system (CRS) says what the numbers of a raster's position mean on the Earth (metres in a projection, or degrees of latitude and longitude). Sibuyan's file has none, so Terraflux takes its cell sides, 250, as metres, which is what they are. With your own maps, check that both dates have the same CRS, extent and cell size; if not, put one on the grid of the other first with raster.match.

The template Land change analysis (menu Templates → Land change) builds the same kind of model on maps drawn by chance; replace its first nodes with raster.read nodes of your files to use it with your own data.

Build the model#

Read the two dates#

A model in Terraflux is a graph: nodes are operations, and the lines join an output of one node to an input of another. Start with the two maps. Add a raster.read node (library: IO), set its path to sibuyan_landuse.tif and its band to 1997; add a second one with the band 2011. A band can be a band's name or its number counted from 0. Each node has an output map: the band as a map of numbers, the sea (255 in the file) as no value.

Then add change.crosstab from the Change detection subject of the library. Drag from the map port of the 1997 node to its from input, and from the 2011 node's map to its to input. The order matters: from gives the rows, to the columns.

format = "terraflux/1"

[[nodes]]
id = "land_1997"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "1997" }
note = "Land use in 1997: 1 forest, 2 coconut, 3 grassland, 4 rice, 5 other."

[[nodes]]
id = "land_2011"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "2011" }
note = "The same classes in 2011."

[[nodes]]
id = "crosstab"
op = "change.crosstab"
inputs = { from = "land_1997.raster", to = "land_2011.raster" }
note = "Rows: the class in 1997; columns: the class in 2011; cells: how many."

Select the crosstab node and press Ctrl+Enter: the model runs up to it and its table appears in the Viewer. It has a row per class of 1997 (from) and a column per class of 2011, the cells in each pair. Classes found in neither map are left out, and so are cells without a value at either date.

Every transition, in km²#

The matrix counts cells. To report areas, and to handle more than two dates, add change.transitions. It takes the maps as one input, maps: a stack (several maps of one grid, a layer each). Add a third raster.read node with the same path and no band: its raster holds both bands, named 1997 and 2011. Connect it to maps.

change.transitions reads the dates from the layers' names (or from its times setting) and a cell's area from the grid (or from cell_area, in m²). It gives four tables: table, a row per pair of classes with the years, the classes, the area in km² and the cells (OpenLand's contingency table; Exavier & Zeilhofer, 2020); one_step, the same from the first date to the last when there are more than two; areas, every class's cells at every date; and matrix, the transition matrix from the first date to the last.

Add change.suggest beside it, on the same stack. It lists every transition with its share of all the change and of its class's area, largest first, and marks as suggested those that make at least least % of the change. Set least to 5.

format = "terraflux/1"

[[nodes]]
id = "land_1997"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "1997" }

[[nodes]]
id = "land_2011"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "2011" }

[[nodes]]
id = "dates"
op = "raster.read"
params = { path = "sibuyan_landuse.tif" }
note = "No band: the stack of both, layers named 1997 and 2011."

[[nodes]]
id = "crosstab"
op = "change.crosstab"
inputs = { from = "land_1997.raster", to = "land_2011.raster" }

[[nodes]]
id = "transitions"
op = "change.transitions"
inputs = { maps = "dates.raster" }
note = "Years from the layers' names; a cell's area from the grid (250 m x 250 m)."

[[nodes]]
id = "suggest"
op = "change.suggest"
inputs = { maps = "dates.raster" }
params = { least = 5 }
note = "Transitions making at least 5% of the change."

The components, and a map of the transitions#

Add change.components. It accepts either two maps (from, to) or a stack (maps, from its first layer to its last); connect the two maps. Its analysis setting names the columns: change (the default) gives gain, persistence and loss; error gives omission, agreement and commission, the words for comparing a map with a reference. Its table has a row per class and an overall row at the end (with an empty class); overall has the total split into quantity, exchange, shift and allocation, in cells.

A table cannot show where the change happened. Add a raster.calc node (library: Map algebra) with two inputs: connect the 1997 map and name the input a, the 2011 map as b (in the Node pane, an input's name is what the expression calls it). Its expression codes every transition with two digits:

0 if a == b else a * 10 + b

so 13 is forest → grassland and 42 rice → coconut, 0 no change. Click the node to see the map in the Viewer, and keep it with the model with Plots › Save as a new plot (titled Transitions 1997-2011 below).

format = "terraflux/1"

[[nodes]]
id = "land_1997"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "1997" }

[[nodes]]
id = "land_2011"
op = "raster.read"
params = { path = "sibuyan_landuse.tif", band = "2011" }

[[nodes]]
id = "dates"
op = "raster.read"
params = { path = "sibuyan_landuse.tif" }

[[nodes]]
id = "crosstab"
op = "change.crosstab"
inputs = { from = "land_1997.raster", to = "land_2011.raster" }

[[nodes]]
id = "transitions"
op = "change.transitions"
inputs = { maps = "dates.raster" }

[[nodes]]
id = "components"
op = "change.components"
inputs = { from = "land_1997.raster", to = "land_2011.raster" }
note = "Gain, persistence, loss; quantity, exchange, shift."

[[nodes]]
id = "suggest"
op = "change.suggest"
inputs = { maps = "dates.raster" }
params = { least = 5 }

[[nodes]]
id = "from_to"
op = "raster.calc"
inputs = { a = "land_1997.raster", b = "land_2011.raster" }
params = { expression = "0 if a == b else a * 10 + b" }
note = "13: forest to grassland, 42: rice to coconut... 0: no change."

[[plots]]
id = "plot"
title = "Transitions 1997-2011"
layers = [{ from = "from_to.map" }]

[outputs.crosstab]
from = "crosstab.table"

[outputs.transitions]
from = "transitions.table"

[outputs.components]
from = "components.table"

[outputs.overall]
from = "components.overall"

terraflux run sibuyan_change.toml --out output

Read the results#

The transition matrix#

The crosstab node gives, in cells of 0.0625 km²:

Read it row by row. Of the 4,723 cells of forest in 1997, 4,160 were still forest in 2011, 320 had become grassland, 126 coconut and 116 rice. The diagonal sums to 6,483 cells: 89.7% of the island persisted and 743 cells (46.4 km², 10.3%) changed. Read the columns too: the forest column has nothing off the diagonal, so not a single cell became forest. Coconut gained from every other class of farmland, and rice lost 61 of its 224 cells to coconut.

Where it happened#

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The map shows two different processes. Forest turned into grassland in large clumps in the north-west and along the inner edge of the eastern plain, where cleared land is left to grass; forest became rice in a strip along the south-west coast. Meanwhile the conversions from grassland and rice to coconut happened within the farmland of the east, at the edges of the plantations. A transition matrix gives the first impression of these processes; the map shows that they happen in different places, which is the first clue to their drivers.

Gains, losses, net change and swap#

The components table, converted to km² (cells × 0.0625):

Forest lost 35.2 km², 11.9% of its area in 1997. Its loss is all net change: it never gained, so its change is pure quantity. Rice tells a different story. Its area grew by only 4 km² (from 14.0 to 18.0 km²), but 12.5 km² of rice changed: it gained 8.25 km² from forest and grassland and lost 4.25 km² to coconut. Two thirds of rice's change is swap, invisible in a comparison of areas. Grassland is similar: its area grew by 15.7 km², but 27.4 km² of grassland was involved in change.

Quantity, exchange and shift#

Overall, the overall table gives 743 cells of change: 563 of quantity (7.8% of the island), 50 of exchange (0.7%) and 130 of shift (1.8%), so 180 cells (2.5%) of allocation. Three quarters of Sibuyan's change was the growth of the farmland at the expense of the forest; a quarter was the farmland rearranging itself. Exchange is small because few pairs of classes traded in both directions: 18 cells went each way between coconut and grassland, and 7 between grassland and rice, so 2 × 18 + 2 × 7 = 50 cells of exchange. Most of the allocation is shift: grassland and rice gained from forest in one place and lost to coconut in another, so they moved towards the forest.

The transitions worth modelling#

change.suggest with least = 5:

Five transitions make 94% of the change. The two columns of percentages answer different questions: forest → grassland is the largest transition (43% of all change) but took only 6.8% of the forest, while rice → coconut is small on the island's scale (8%) yet took 27% of the rice. A model of the island's future would model the first five; the others have too few cells to fit anything to.

Settings that change the answer#

The operators of this tutorial have few settings, and none of them changes the counts: times and cell_area only convert cells into years and km². What changes the answer is the data: the legend (merge coconut and rice into "farmland" and their exchange disappears), the method that made the maps, and the resolution.

The slider shows the same analysis on the two maps made coarser, each new cell taking the most common class of the cells it covers (raster.resample with resampling = "mode"), from 250 m to 1 km:

format = "terraflux/1"

[[nodes]]
id = "dates"
op = "raster.read"
params = { path = "sibuyan_landuse.tif" }

[[nodes]]
id = "coarser"
op = "raster.resample"
inputs = { raster = "dates.raster" }
params = { resolution = 750, resampling = "mode" }
note = "Each new cell takes the most common class of those it covers."

[[nodes]]
id = "components"
op = "change.components"
inputs = { maps = "coarser.raster" }

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PlayCell size (m)

Two things happen. Small transitions vanish: the thin strips of forest → rice along the coast and the scattered cells of coconut gains are absorbed by their neighbours, and at 1 km the shift falls from 1.8% to 0.8%. And the change does not shrink steadily: at 1 km the quantity grows, to 9.4%, because the majority rule also redraws the coast and turns mixed cells into whichever class happens to dominate them at each date. Coarsening by majority is not a neutral operation. If maps must be compared at a coarser resolution, the multi-resolution methods of part 3 do it without redrawing the classes.

Common pitfalls#

• Maps that are not on the same grid. A cross-tabulation pairs cell k of one map with cell k of the other. If the grids differ by half a cell, every boundary becomes false change. Terraflux refuses maps of different sizes, but it cannot know whether two grids of the same size cover the same ground: check the extents.
• No-data codes counted as a class. If 255 (or 0) means "no data" but the file does not say so, it becomes a class with its own row and column. Look at the classes of the matrix; set nodata on raster.read when the file does not declare it.
• Rows and columns swapped. Rows are from (the earlier map, or the map being compared), columns to (the later map, or the reference). Swapping them turns gains into losses.
• Areas in degrees. change.transitions takes a cell's area from the grid's units: on a map in latitude and longitude that is square degrees, and the km² are nonsense. Reproject the maps (raster.reproject) or give cell_area in m².
• Percent of what? "Forest lost 12%" (of the forest) and "forest loss was 7.8% of the island" are both true here. Say which denominator you use.
• Map error taken for change. Two independent maps, each 90% correct, may disagree on up to 20% of the cells over a landscape that did not change. Use maps made the same way, check them against reference data, and estimate the areas of change from a sample (Olofsson et al., 2014; see part 3).
• Swap read as no change. Two maps with the same areas of each class can still differ everywhere. Report the swap, not only the net change.

Going further#

• Part 2: intensity analysis: whether a period changed faster than another, which classes gained and lost actively, and which transitions targeted a class.
• Part 3: comparing maps: quantity and allocation disagreement, differences at several resolutions, kappa and the estimation of areas from a sample.
• Part 4: forest gain and loss: GuidosToolbox's simple and morphological change, and the change of fragmentation.
• Land change simulation: the transitions suggested here are what a simulation model is calibrated on.
• Operators: change.crosstab, change.transitions, change.components, change.suggest, change.count (how many times each cell changed along a series of maps).

References#

• Exavier, R., & Zeilhofer, P. (2020). OpenLand: software for quantitative analysis and visualization of land use and cover change. The R Journal, 12(2), 359. https://doi.org/10.32614/RJ-2021-021
• Olofsson, P., Foody, G. M., Herold, M., Stehman, S. V., Woodcock, C. E., & Wulder, M. A. (2014). Good practices for estimating area and assessing accuracy of land change. Remote Sensing of Environment, 148, 42–57. https://doi.org/10.1016/j.rse.2014.02.015
• Pontius, R. G., Jr., & Santacruz, A. (2014). Quantity, exchange, and shift components of difference in a square contingency table. International Journal of Remote Sensing, 35(21), 7543–7554. https://doi.org/10.1080/2150704X.2014.969814
• Pontius, R. G., Jr., Shusas, E., & McEachern, M. (2004). Detecting important categorical land changes while accounting for persistence. Agriculture, Ecosystems & Environment, 101(2–3), 251–268. https://doi.org/10.1016/j.agee.2003.09.008
• Verburg, P. H., Soepboer, W., Veldkamp, A., Limpiada, R., Espaldon, V., & Mastura, S. S. A. (2002). Modeling the spatial dynamics of regional land use: the CLUE-S model. Environmental Management, 30(3), 391–405. https://doi.org/10.1007/s00267-002-2630-x